reserve X,Y,Z for set,
  a,b,c,d,x,y,z,u for object,
  R for Relation,
  A,B,C for Ordinal;
reserve H for Function;
reserve f,g for Function;
reserve M for non empty set;

theorem
  RelIncl {x} = {[x,x]}
proof
A1: for Y,Z being set st Y in {x} & Z in {x} holds [Y,Z] in {[x,x]} iff Y c= Z
  proof
    let Y,Z be set;
    assume that
A2: Y in {x} and
A3: Z in {x};
A4: Y = x by A2,TARSKI:def 1;
    hence [Y,Z] in {[x,x]} implies Y c= Z by A3,TARSKI:def 1;
    Z = x by A3,TARSKI:def 1;
    hence thesis by A4,TARSKI:def 1;
  end;
  field {[x,x]} = {x} by RELAT_1:173;
  hence thesis by A1,Def1;
end;
